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Worksheet: Rotation and Reflection Matrices

In this worksheet, we will practice finding the matrix of linear transformation which rotates every vector by a given angle and reflects them in the x- or y-axis.

Q1:

Find the matrix with respect to the standard basis vectors for the linear transformation that rotates every vector in ℝ 2 through an angle of πœ‹ 3 and then reflects it across the 𝑦 -axis.

  • A ⎑ ⎒ ⎒ ⎒ ⎣ 1 2 √ 3 2 √ 3 2 1 2 ⎀ βŽ₯ βŽ₯ βŽ₯ ⎦
  • B ⎑ ⎒ ⎒ ⎒ ⎣ βˆ’ √ 3 2 1 2 √ 3 2 1 2 ⎀ βŽ₯ βŽ₯ βŽ₯ ⎦
  • C ⎑ ⎒ ⎒ ⎒ ⎣ 1 2 βˆ’ √ 3 2 √ 3 2 1 2 ⎀ βŽ₯ βŽ₯ βŽ₯ ⎦
  • D ⎑ ⎒ ⎒ ⎒ ⎣ βˆ’ 1 2 √ 3 2 √ 3 2 1 2 ⎀ βŽ₯ βŽ₯ βŽ₯ ⎦
  • E ⎑ ⎒ ⎒ ⎒ ⎣ βˆ’ √ 3 2 1 2 1 2 √ 3 2 ⎀ βŽ₯ βŽ₯ βŽ₯ ⎦

Q2:

A linear transformation is formed by reflecting every vector in in the -axis and then rotating the resulting vector through an angle of . Find the matrix of this linear transformation.

  • A
  • B
  • C
  • D
  • E

Q3:

Let the matrix 𝐴 represent rotation in the plane through an angle of πœƒ and let the matrix 𝐡 represent reflection in the π‘₯ -axis.

What is the matrix 𝐴 ?

  • A  πœƒ βˆ’ πœƒ πœƒ πœƒ  s i n c o s c o s s i n
  • B  πœƒ πœƒ πœƒ πœƒ  c o s s i n s i n c o s
  • C  πœƒ πœƒ πœƒ πœƒ  s i n c o s c o s s i n
  • D  πœƒ βˆ’ πœƒ πœƒ πœƒ  c o s s i n s i n c o s
  • E  πœƒ πœƒ βˆ’ πœƒ πœƒ  c o s s i n s i n c o s

What is the matrix 𝐡 ?

  • A  1 0 0 βˆ’ 1 
  • B  0 1 1 0 
  • C  βˆ’ 1 0 0 βˆ’ 1 
  • D  βˆ’ 1 0 0 1 
  • E  0 βˆ’ 1 βˆ’ 1 0 

What is the matrix 𝐴 𝐡 ?

  • A  πœƒ βˆ’ πœƒ πœƒ πœƒ  s i n c o s c o s s i n
  • B  πœƒ πœƒ βˆ’ πœƒ πœƒ  c o s s i n s i n c o s
  • C  πœƒ πœƒ πœƒ βˆ’ πœƒ  c o s s i n s i n c o s
  • D  πœƒ βˆ’ πœƒ πœƒ πœƒ  c o s s i n s i n c o s
  • E  πœƒ πœƒ πœƒ βˆ’ πœƒ  s i n c o s c o s s i n

Q4:

Suppose that the matrix represents rotation about the origin through an angle of (measuring between and ) and the matrix represents reflection in the -axis.

Find .

  • A
  • B
  • C
  • D
  • E

Note that is a reflection in a line through the origin. Let this line of reflection have equation . By considering the image of the vector , determine the measure of the angle between the -axis and the line of reflection.

  • A
  • B
  • C
  • D
  • E

What, therefore, is the slope of the line of reflection?

  • A
  • B
  • C
  • D
  • E

If lies in the direction of the line of reflection, what is ?

  • A
  • B
  • C
  • D
  • E

By solving the equation obtained in the previous part, find another expression for the slope of the line of reflection.

  • A
  • B
  • C
  • D
  • E

Q5:

A linear transformation is formed by rotating every vector in through an angle of counterclockwise (when viewed from the positive -axis) about the -axis and then reflecting the resulting vector in the -plane. Find the matrix of this linear transformation.

  • A
  • B
  • C
  • D
  • E

Q6:

A linear transformation is formed by rotating every vector in through an angle of and then reflecting the resulting vector in the -axis. Find the matrix of this linear transformation.

  • A
  • B
  • C
  • D
  • E

Q7:

A linear transformation is formed by reflecting every vector in across the -axis and then rotating the resulting vector through an angle of . Find the matrix of this linear transformation.

  • A
  • B
  • C
  • D
  • E

Q8:

A linear transformation is formed by rotating every vector in through an angle of , reflecting the resulting vector in the -axis, and finally reflecting this vector in the -axis. Find the matrix of this linear transformation.

  • A
  • B
  • C
  • D
  • E

Q9:

Consider the given figure.

The points 𝑂 ( 0 , 0 ) , 𝐴 ( 1 , 0 ) , 𝐡 ( 1 , 1 ) , and 𝐢 ( 0 , 1 ) are corners of the unit square. This square is reflected in the line 𝑂 𝐷 with equation 𝑦 = π‘˜ π‘₯ to form the image 𝑂 𝐴 𝐡 𝐢 βˆ— βˆ— βˆ— .

As 𝐴 βˆ— is the image of 𝐴 in the line through 𝑂 and 𝐷 , π‘š ∠ 𝐴 𝑂 𝐴 = 2 π‘š ∠ 𝐷 𝑂 𝐴 βˆ— . Use this fact and the identity t a n t a n t a n 2 πœƒ = 2 πœƒ 1 βˆ’ πœƒ  to find the gradient and hence equation of βƒ–       βƒ— 𝑂 𝐴 βˆ— from the gradient of βƒ–      βƒ— 𝑂 𝐷 .

  • A 𝑦 = π‘˜ 1 βˆ’ π‘˜ π‘₯ 
  • B 𝑦 = 2 π‘˜ π‘˜ βˆ’ 1 π‘₯ 
  • C 𝑦 = π‘˜ π‘˜ βˆ’ 1 π‘₯ 
  • D 𝑦 = 2 π‘˜ 1 βˆ’ π‘˜ π‘₯ 
  • E 𝑦 = 2 π‘˜ 1 + π‘˜ π‘₯ 

Using the fact that βƒ–        βƒ— 𝑂 𝐢 βˆ— is perpendicular to βƒ–       βƒ— 𝑂 𝐴 βˆ— , find the equation of βƒ–        βƒ— 𝑂 𝐢 βˆ— .

  • A 𝑦 = π‘˜ βˆ’ 1 2 π‘˜ π‘₯ 
  • B 𝑦 = 2 π‘˜ π‘˜ βˆ’ 1 π‘₯ 
  • C 𝑦 = 2 π‘˜ 1 βˆ’ π‘˜ π‘₯ 
  • D 𝑦 = 1 βˆ’ π‘˜ 2 π‘˜ π‘₯ 
  • E 𝑦 = π‘˜ βˆ’ 1 2 π‘˜ π‘₯ 

Using the fact that 𝑂 𝐢 = 𝑂 𝐴 = 1 βˆ— βˆ— , find the coordinates of 𝐢 βˆ— and 𝐴 βˆ— .

  • A 𝐢 = ο€Ύ π‘˜ 1 + π‘˜ , π‘˜ βˆ’ 1 1 + π‘˜  βˆ—    , 𝐴 = ο€Ύ 1 βˆ’ π‘˜ 1 + π‘˜ , π‘˜ 1 + π‘˜  βˆ—   
  • B 𝐢 = ο€Ύ 2 π‘˜ 1 + π‘˜ , π‘˜ βˆ’ 1 1 + π‘˜  βˆ—  , 𝐴 = ο€Ύ 1 βˆ’ π‘˜ 1 + π‘˜ , 2 π‘˜ 1 + π‘˜  βˆ— 
  • C 𝐢 = ο€Ύ 2 π‘˜ 1 + π‘˜ , π‘˜ βˆ’ 1 1 + π‘˜  βˆ—    , 𝐴 = ο€Ύ 1 βˆ’ π‘˜ 1 + π‘˜ , 2 π‘˜ 1 + π‘˜  βˆ—   
  • D 𝐢 = ο€Ύ π‘˜ βˆ’ 1 1 + π‘˜ , 2 π‘˜ 1 + π‘˜  βˆ—    , 𝐴 = ο€Ύ 2 π‘˜ 1 + π‘˜ , 1 βˆ’ π‘˜ 1 + π‘˜  βˆ—   
  • E 𝐢 = ο€Ύ π‘˜ βˆ’ 1 1 + π‘˜ , 2 π‘˜ 1 + π‘˜  βˆ—  , 𝐴 = ο€Ύ 2 π‘˜ 1 + π‘˜ , 1 βˆ’ π‘˜ 1 + π‘˜  βˆ— 

Using the fact that a reflection in a line through the origin is a linear transformation, find the matrix which represents reflection in the line 𝑦 = π‘˜ π‘₯ .

  • A ⎑ ⎒ ⎒ ⎒ ⎣ 1 βˆ’ π‘˜ 1 + π‘˜ 2 π‘˜ 1 + π‘˜ 2 π‘˜ 1 + π‘˜ π‘˜ βˆ’ 1 1 + π‘˜ ⎀ βŽ₯ βŽ₯ βŽ₯ ⎦      
  • B ⎑ ⎒ ⎒ ⎒ ⎣ 1 + π‘˜ 1 βˆ’ π‘˜ 1 + π‘˜ 2 π‘˜ 1 + π‘˜ 2 π‘˜ 1 + π‘˜ π‘˜ βˆ’ 1 ⎀ βŽ₯ βŽ₯ βŽ₯ ⎦      
  • C ⎑ ⎒ ⎒ ⎒ ⎣ 1 βˆ’ π‘˜ 1 + π‘˜ 2 π‘˜ 1 + π‘˜ βˆ’ 2 π‘˜ 1 + π‘˜ 1 βˆ’ π‘˜ 1 + π‘˜ ⎀ βŽ₯ βŽ₯ βŽ₯ ⎦      
  • D ⎑ ⎒ ⎒ ⎒ ⎣ 1 βˆ’ π‘˜ 1 + π‘˜ 2 π‘˜ 1 + π‘˜ 2 π‘˜ 1 + π‘˜ π‘˜ βˆ’ 1 1 + π‘˜ ⎀ βŽ₯ βŽ₯ βŽ₯ ⎦  
  • E ⎑ ⎒ ⎒ ⎒ ⎣ 1 βˆ’ π‘˜ 1 + π‘˜ 2 π‘˜ 1 + π‘˜ 2 π‘˜ 1 + π‘˜ 1 βˆ’ π‘˜ 1 + π‘˜ ⎀ βŽ₯ βŽ₯ βŽ₯ ⎦      

Q10:

A linear transformation is formed by rotating every vector in through an angle of and then reflecting the resulting vector in the -axis. Find the matrix of this linear transformation.

  • A
  • B
  • C
  • D
  • E

Q11:

A linear transformation is formed by reflecting every vector in ℝ 2 in the π‘₯ -axis and then rotating the resulting vector through an angle of πœ‹ 6 . Find the matrix of this linear transformation.

  • A ⎑ ⎒ ⎒ ⎒ ⎣ √ 3 2 βˆ’ 1 2 βˆ’ 1 2 βˆ’ √ 3 2 ⎀ βŽ₯ βŽ₯ βŽ₯ ⎦
  • B ⎑ ⎒ ⎒ ⎒ ⎣ √ 3 2 βˆ’ 1 2 1 2 βˆ’ √ 3 2 ⎀ βŽ₯ βŽ₯ βŽ₯ ⎦
  • C ⎑ ⎒ ⎒ ⎒ ⎣ √ 3 2 βˆ’ 1 2 1 2 √ 3 2 ⎀ βŽ₯ βŽ₯ βŽ₯ ⎦
  • D ⎑ ⎒ ⎒ ⎒ ⎣ √ 3 2 1 2 1 2 βˆ’ √ 3 2 ⎀ βŽ₯ βŽ₯ βŽ₯ ⎦
  • E ⎑ ⎒ ⎒ ⎒ ⎣ √ 3 2 0 0 βˆ’ √ 3 2 ⎀ βŽ₯ βŽ₯ βŽ₯ ⎦

Q12:

A linear transformation is formed by reflecting every vector in in the -axis and then rotating the resulting vector through an angle of . Find the matrix of this linear transformation.

  • A
  • B
  • C
  • D
  • E

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